Invertibility via distance for noncentered random matrices with continuous distributions

@article{Tikhomirov2020InvertibilityVD,
  title={Invertibility via distance for noncentered random matrices with continuous distributions},
  author={Konstantin E. Tikhomirov},
  journal={Random Struct. Algorithms},
  year={2020},
  volume={57},
  pages={526-562}
}
Let $A$ be an $n\times n$ random matrix with independent rows $R_1(A),\dots,R_n(A)$, and assume that for any $i\leq n$ and any three-dimensional linear subspace $F\subset {\mathbb R}^n$ the orthogonal projection of $R_i(A)$ onto $F$ has distribution density $\rho(x):F\to{\mathbb R}_+$ satisfying $\rho(x)\leq C_1/\max(1,\|x\|_2^{2000})$ ($x\in F$) for some constant $C_1>0$. We show that for any fixed $n\times n$ real matrix $M$ we have $${\mathbb P}\{s_{\min}(A+M)\leq t n^{-1/2}\}\leq C'\, t… Expand
Overlaps, Eigenvalue Gaps, and Pseudospectrum under real Ginibre and Absolutely Continuous Perturbations
TLDR
A key ingredient in the proof is new lower tail bounds on the small singular values of the complex shifts $z-(A+\gamma G_n)$ which recover the tail behavior of thecomplex Ginibre ensemble when $\Im z\neq 0$ yields sharp control on the area of the pseudospectrum $\Lambda_\epsilon(A+ \gammaG_n). Expand
On the smoothed analysis of the smallest singular value with discrete noise
TLDR
It is shown that any bound of the form s_n(A+M) = O(\epsilon) + 2e^{-\Omega(n)], provided only that $A$ has $\Omega (n)$ singular values which are $O(\sqrt{n})$. Expand
A remark on the smallest singular value of powers of Gaussian matrices
Let $n,k\geq 1$ and let $G$ be the $n\times n$ random matrix with i.i.d. standard real Gaussian entries. We show that there are constants $c_k,C_k>0$ depending only on $k$ such that the smallestExpand
On the real Davies' conjecture
TLDR
It is proved that, with high probability, taking $E$ to be a sufficiently small multiple of an i.i.d. real sub-Gaussian matrix of bounded density suffices, and non-asymptotic estimates on the minimum possible distance between any two eigenvalues of a random matrix whose entries have arbitrary means are proved. Expand
On delocalization of eigenvectors of random non-Hermitian matrices
We study delocalization of null vectors and eigenvectors of random matrices with i.i.d entries. Let A be an $$n\times n$$ n × n random matrix with i.i.d real subgaussian entries of zero mean and unitExpand
Anti-concentration for subgraph counts in random graphs
Fix a graph $H$ and some $p\in (0,1)$, and let $X_H$ be the number of copies of $H$ in a random graph $G(n,p)$. Random variables of this form have been intensively studied since the foundational workExpand
Universality of the least singular value for the sum of random matrices
We consider the least singular value of $M = R^* X T + U^* YV$, where $R,T,U,V$ are independent Haar-distributed unitary matrices and $X, Y$ are deterministic diagonal matrices. Under weak conditionsExpand
Least singular value and condition number of a square random matrix with i.i.d. rows
We consider a square random matrix made by i.i.d. rows with any distribution and prove that, for any given dimension, the probability for the least singular value to be in [0; $\epsilon$) is at leastExpand
Matrix anti-concentration inequalities with applications
  • Zipei Nie
  • Computer Science, Mathematics
  • ArXiv
  • 2021
TLDR
A better singular value bound is proved for the Krylov space matrix, which leads to a faster and simpler algorithm for solving sparse linear systems and establishes two matrix anti-concentration inequalities. Expand
The smallest singular value of heavy-tailed not necessarily i.i.d. random matrices via random rounding
We are concerned with the small ball behavior of the smallest singular value of random matrices. Often, establishing such results involves, in some capacity, a discretization of the unit sphere. ThisExpand
...
1
2
...

References

SHOWING 1-10 OF 36 REFERENCES
Invertibility of Sparse non-Hermitian matrices
We consider a class of sparse random matrices of the form $A_n =(\xi_{i,j}\delta_{i,j})_{i,j=1}^n$, where $\{\xi_{i,j}\}$ are i.i.d.~centered random variables, and $\{\delta_{i,j}\}$ areExpand
The circular law for sparse non-Hermitian matrices
For a class of sparse random matrices of the form $A_n =(\xi_{i,j}\delta_{i,j})_{i,j=1}^n$, where $\{\xi_{i,j}\}$ are i.i.d.~centered sub-Gaussian random variables of unit variance, andExpand
Small Ball Probabilities for Linear Images of High-Dimensional Distributions
Author(s): Rudelson, Mark; Vershynin, Roman | Abstract: We study concentration properties of random vectors of the form $AX$, where $X = (X_1, ..., X_n)$ has independent coordinates and $A$ is aExpand
Lower bounds for the smallest singular value of structured random matrices
We obtain lower tail estimates for the smallest singular value of random matrices with independent but non-identically distributed entries. Specifically, we consider $n\times n$ matrices with complexExpand
Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries
Let A = (aij) be an n × n random matrix with i.i.d. entries such that Ea11 = 0 and Ea112 = 1. We prove that for any δ > 0 there is L > 0 depending only on δ, and a subset N of B2n of cardinality atExpand
Concentration of mass on convex bodies
Abstract.We establish sharp concentration of mass inequality for isotropic convex bodies: there exists an absolute constant c >  0 such that if K is an isotropic convex body in $$\mathbb{R}^{n}$$,Expand
Matrix regularizing effects of Gaussian perturbations
The addition of noise has a regularizing effect on Hermitian matrices. This effect is studied here for $H=A+V$, where $A$ is the base matrix and $V$ is sampled from the GOE or the GUE random matrixExpand
Inverse Littlewood-Offord theorems and the condition number of random discrete matrices
Consider a random sum r)\V\ + • • • + r]nvn, where 771, . . . , rin are independently and identically distributed (i.i.d.) random signs and vi, . . . , vn are integers. The Littlewood-Offord problemExpand
Random Matrices: the Distribution of the Smallest Singular Values
Let ξ be a real-valued random variable of mean zero and variance 1. Let Mn(ξ) denote the n × n random matrix whose entries are iid copies of ξ and σn(Mn(ξ)) denote the least singular value of Mn(ξ).Expand
On the probability that a random ±1-matrix is singular
We report some progress on the old problem of estimating the probability, Pn, that a random n× n ± 1 matrix is singular: Theorem. There is a positive constant ε for which Pn < (1− ε)n. This is aExpand
...
1
2
3
4
...